Central splitting of manifolds with no conjugate points

Central conjugate locus of 2-step nilpotent Lie groups

The goals of this article are twofold : 1) to compute the conjugate locus of a geodesic that lies in the center of a simply connected, 2-step nilpotent Lie group with a left invariant metric 2) compare the isometry types of two such nilpotent Lie groups whose conjugate loci for central geodesics are "the same" in a suitable sense. The first goal is achieved. The second is elusive, but we obtain a partial result.

没有共轭点度量的Σ×S¹流形等距分割

Isometric Splitting of Metrics Without Conjugate Points on $Σ\times S^1$

研究没有共轭点的Σ×S¹流形上的黎曼度量，证明其必须为黎曼积度量，并探讨其几何和动力学后果。

Manifolds without conjugate points and their fundamental groups

We show that in the fundamental groups of closed manifolds without conjugate points centralizers of all elements virtually split.

Smooth globally hyperbolic splittings and temporal functions

Geroch's theorem about the splitting of globally hyperbolic spacetimes is a central result in global Lorentzian Geometry. Nevertheless, this result was obtained at a topological level, and the possibility to obtain a metric (or, at least, smooth) version has been controversial since its publication in 1970. In fact, this problem has remained open until a definitive proof, recently provided by the authors. Our purpose is to summarize the history of the problem, explain the smooth and metric splitting results (including smoothability of time functions in stably causal spacetimes), and sketch the…